This is the content of Theorem REF  and for us it is a simple consequence of the Cauchy identities (REF ) and their generalizations to arbitrary specializations. Using different methods, the result was first proved by Johansson [1]}, and reproved subsequently by Basor–Ehrhardt, Dehaye and Deift–Its–Krasovsky in [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
